A "whispering
room" is one with an elliptically-arched ceiling. If someone stands
at one focus of the ellipse and whispers something to his friend, the
dispersed sound waves are reflected by the ceiling and concentrated
at the other focus, allowing people across the room to clearly hear
what he said. Suppose such gallery has a ceiling reaching twenty feet
above the five-foot-high vertical walls at its tallest point (so the
cross-section is half an ellipse topping two vertical lines at either
end), and suppose the foci of the ellipse are thirty feet apart. What
is the height of the ceiling above each "whispering point"?

Since the ceiling is half of an ellipse
(the top half, specifically), and since the foci will be on a line between
the tops of the "straight" parts of the side walls, the foci
will be five feet above the floor, which sounds about right for people
talking and listening: five feet high is close to face-high on most
adults.

I'll center my ellipse above the origin,
so (h, k)
= (0, 5). The foci are thirty feet
apart, so they're 15
units to either side of the center. In particular, c
= 15. Since the elliptical part of
the room's cross-section is twenty feet high above the center, and since
this "shorter" direction is the semi-minor axis, then b
= 20. The equation b2
= a2 – c2
gives me 400 = a2
– 225, so a2
= 625. Then the equation for the
elliptical ceiling is:

I need to find the height of the ceiling
above the foci. I prefer positive numbers, so I'll look at the focus
to the right of the center. The height (from the ellipse's central line
through its foci, up to the ceiling) will be the y-value
of the ellipse when x
= 15:

Satellites can be put
into elliptical orbits if they need only sometimes to be in high- or
low-earth orbit, thus avoiding the need for propulsion and navigation
in low-earth orbit and the expense of launching into high-earth orbit.
Suppose a satellite is in an elliptical orbit, with a
= 4420 and b
= 4416, and
with the center of the Earth being at one of the foci of the ellipse.
Assuming the Earth has a radius of about 3960
miles, find the lowest and highest altitudes of the satellite above
the Earth.

The lowest altitude will be at the vertex
closer to the Earth; the highest altitude will be at the other vertex.
Since I need to measure these altitudes from the focus, I need to find
the value of c.

b2 = a2
– c2c2 = a2
– b2 = 44202 – 44162 = 35,344

Then c
= 188. If I set the center of my
ellipse at the origin and make this a wider-than-tall ellipse, then
I can put the Earth's center at the point (188,
0).

(This means, by the way, that there isn't
much difference between the circumference of the Earth and the path of
the satellite. The center of the elliptical orbit is actually inside the
Earth, and the ellipse, having an eccentricity of e
= 188 / 4420, or about 0.04,
is pretty close to being a circle.)

The vertex closer to the end of the ellipse
containing the Earth's center will be at 4420
units from the ellipse's center, or 4420
– 188 = 4232 units from the center
of the Earth. Since the Earth's radius is 3960
units, then the altitude is 4232
– 3960 = 272. The other vertex is
4420 + 188 = 4608
units from the Earth's center, giving me an altitude of 4608
– 3960 = 648 units.

The minimum altitude
is 272
miles above the Earth; the maximum altitude is 648
miles above the Earth.